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Theorem isoval 13683
Description: The inverse relation is a function, which is to say that every morphism has at most one inverse. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
invfval.b  |-  B  =  ( Base `  C
)
invfval.n  |-  N  =  (Inv `  C )
invfval.c  |-  ( ph  ->  C  e.  Cat )
invfval.x  |-  ( ph  ->  X  e.  B )
invfval.y  |-  ( ph  ->  Y  e.  B )
isoval.n  |-  I  =  (  Iso  `  C
)
Assertion
Ref Expression
isoval  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )

Proof of Theorem isoval
Dummy variables  x  c  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isoval.n . . . 4  |-  I  =  (  Iso  `  C
)
2 invfval.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
3 fveq2 5541 . . . . . . . 8  |-  ( c  =  C  ->  (Inv `  c )  =  (Inv
`  C ) )
4 invfval.n . . . . . . . 8  |-  N  =  (Inv `  C )
53, 4syl6eqr 2346 . . . . . . 7  |-  ( c  =  C  ->  (Inv `  c )  =  N )
65coeq2d 4862 . . . . . 6  |-  ( c  =  C  ->  (
( z  e.  _V  |->  dom  z )  o.  (Inv `  c ) )  =  ( ( z  e. 
_V  |->  dom  z )  o.  N ) )
7 df-iso 13668 . . . . . 6  |-  Iso  =  ( c  e.  Cat  |->  ( ( z  e. 
_V  |->  dom  z )  o.  (Inv `  c )
) )
8 funmpt 5306 . . . . . . 7  |-  Fun  (
z  e.  _V  |->  dom  z )
9 fvex 5555 . . . . . . . 8  |-  (Inv `  C )  e.  _V
104, 9eqeltri 2366 . . . . . . 7  |-  N  e. 
_V
11 cofunexg 5755 . . . . . . 7  |-  ( ( Fun  ( z  e. 
_V  |->  dom  z )  /\  N  e.  _V )  ->  ( ( z  e.  _V  |->  dom  z
)  o.  N )  e.  _V )
128, 10, 11mp2an 653 . . . . . 6  |-  ( ( z  e.  _V  |->  dom  z )  o.  N
)  e.  _V
136, 7, 12fvmpt 5618 . . . . 5  |-  ( C  e.  Cat  ->  (  Iso  `  C )  =  ( ( z  e. 
_V  |->  dom  z )  o.  N ) )
142, 13syl 15 . . . 4  |-  ( ph  ->  (  Iso  `  C
)  =  ( ( z  e.  _V  |->  dom  z )  o.  N
) )
151, 14syl5eq 2340 . . 3  |-  ( ph  ->  I  =  ( ( z  e.  _V  |->  dom  z )  o.  N
) )
1615oveqd 5891 . 2  |-  ( ph  ->  ( X I Y )  =  ( X ( ( z  e. 
_V  |->  dom  z )  o.  N ) Y ) )
17 eqid 2296 . . . . . 6  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) )  =  ( x  e.  B , 
y  e.  B  |->  ( ( x (Sect `  C ) y )  i^i  `' ( y (Sect `  C )
x ) ) )
18 ovex 5899 . . . . . . 7  |-  ( x (Sect `  C )
y )  e.  _V
1918inex1 4171 . . . . . 6  |-  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) )  e.  _V
2017, 19fnmpt2i 6209 . . . . 5  |-  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) )  Fn  ( B  X.  B )
21 invfval.b . . . . . . 7  |-  B  =  ( Base `  C
)
22 invfval.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
23 invfval.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
24 eqid 2296 . . . . . . 7  |-  (Sect `  C )  =  (Sect `  C )
2521, 4, 2, 22, 23, 24invffval 13676 . . . . . 6  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C
) y )  i^i  `' ( y (Sect `  C ) x ) ) ) )
2625fneq1d 5351 . . . . 5  |-  ( ph  ->  ( N  Fn  ( B  X.  B )  <->  ( x  e.  B ,  y  e.  B  |->  ( ( x (Sect `  C )
y )  i^i  `' ( y (Sect `  C ) x ) ) )  Fn  ( B  X.  B ) ) )
2720, 26mpbiri 224 . . . 4  |-  ( ph  ->  N  Fn  ( B  X.  B ) )
28 opelxpi 4737 . . . . 5  |-  ( ( X  e.  B  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
2922, 23, 28syl2anc 642 . . . 4  |-  ( ph  -> 
<. X ,  Y >.  e.  ( B  X.  B
) )
30 fvco2 5610 . . . 4  |-  ( ( N  Fn  ( B  X.  B )  /\  <. X ,  Y >.  e.  ( B  X.  B
) )  ->  (
( ( z  e. 
_V  |->  dom  z )  o.  N ) `  <. X ,  Y >. )  =  ( ( z  e.  _V  |->  dom  z
) `  ( N `  <. X ,  Y >. ) ) )
3127, 29, 30syl2anc 642 . . 3  |-  ( ph  ->  ( ( ( z  e.  _V  |->  dom  z
)  o.  N ) `
 <. X ,  Y >. )  =  ( ( z  e.  _V  |->  dom  z ) `  ( N `  <. X ,  Y >. ) ) )
32 df-ov 5877 . . 3  |-  ( X ( ( z  e. 
_V  |->  dom  z )  o.  N ) Y )  =  ( ( ( z  e.  _V  |->  dom  z )  o.  N
) `  <. X ,  Y >. )
33 ovex 5899 . . . . 5  |-  ( X N Y )  e. 
_V
34 dmeq 4895 . . . . . 6  |-  ( z  =  ( X N Y )  ->  dom  z  =  dom  ( X N Y ) )
35 eqid 2296 . . . . . 6  |-  ( z  e.  _V  |->  dom  z
)  =  ( z  e.  _V  |->  dom  z
)
3633dmex 4957 . . . . . 6  |-  dom  ( X N Y )  e. 
_V
3734, 35, 36fvmpt 5618 . . . . 5  |-  ( ( X N Y )  e.  _V  ->  (
( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  dom  ( X N Y ) )
3833, 37ax-mp 8 . . . 4  |-  ( ( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  dom  ( X N Y )
39 df-ov 5877 . . . . 5  |-  ( X N Y )  =  ( N `  <. X ,  Y >. )
4039fveq2i 5544 . . . 4  |-  ( ( z  e.  _V  |->  dom  z ) `  ( X N Y ) )  =  ( ( z  e.  _V  |->  dom  z
) `  ( N `  <. X ,  Y >. ) )
4138, 40eqtr3i 2318 . . 3  |-  dom  ( X N Y )  =  ( ( z  e. 
_V  |->  dom  z ) `  ( N `  <. X ,  Y >. )
)
4231, 32, 413eqtr4g 2353 . 2  |-  ( ph  ->  ( X ( ( z  e.  _V  |->  dom  z )  o.  N
) Y )  =  dom  ( X N Y ) )
4316, 42eqtrd 2328 1  |-  ( ph  ->  ( X I Y )  =  dom  ( X N Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   <.cop 3656    e. cmpt 4093    X. cxp 4703   `'ccnv 4704   dom cdm 4705    o. ccom 4709   Fun wfun 5265    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   Basecbs 13164   Catccat 13582  Sectcsect 13663  Invcinv 13664    Iso ciso 13665
This theorem is referenced by:  inviso1  13684  invf  13686  invco  13689  isohom  13690  oppciso  13695  funciso  13764  ffthiso  13819  fuciso  13865  setciso  13939  catciso  13955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-inv 13667  df-iso 13668
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